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In this paper we study the semigroup$\mathscr{I}_{\infty}^{\nearrow}(\mathbb{N})$ of partial cofinal monotonebijective transformations of the set of positive integers $\mathbb{N}$. We showthat the semigroup $\mathscr{I}_{\infty}^{\nearrow}(\mathbb{N})$ has algebraicproperties similar to the bicyclic semigroup: it is bisimple and all of itsnon-trivial group homomorphisms are either isomorphisms or group homomorphisms.We also prove that every locally compact topology $\tau$ on$\mathscr{I}_{\infty}^{\nearrow}(\mathbb{N})$ such that$(\mathscr{I}_{\infty}^{\nearrow}(\mathbb{N}),\tau)$ is a topological inversesemigroup, is discrete. Finally, we describe the closure of$(\mathscr{I}_{\infty}^{\nearrow}(\mathbb{N}),\tau)$ in a topologicalsemigroup.

Topological monoids of monotone injective partial selfmaps of $\mathbb{N}$ with cofinite domain and image